arXiv subjects

Richard Santiago

Publications and source records attributed to Richard Santiago.

11 recordsLinked to original sources

arXiv2023

Constant-Competitiveness for Random Assignment Matroid Secretary Without Knowing the Matroid

The Matroid Secretary Conjecture is a notorious open problem in online optimization. It claims the existence of an $O(1)$-competitive algorithm for the Matroid Secretary Problem (MSP). Here, the elements of a weighted matroid appear one-by-one, revealing their weight at appearance, and the task is to select elements online with the goal to get an independent set of largest possible weight. $O(1)$-competitive MSP algorithms have so far only been obtained for restricted matroid classes and for MSP variations, including Random-Assignment MSP (RA-MSP), where an adversary fixes a number of weights equal to the ground set size of the matroid, which then get assigned randomly to the elements of the ground set. Unfortunately, these approaches heavily rely on knowing the full matroid upfront. This is an arguably undesirable requirement, and there are good reasons to believe that an approach towards resolving the MSP Conjecture should not rely on it. Thus, both Soto [SIAM Journal on Computing 2013] and Oveis Gharan & Vondrak [Algorithmica 2013] raised as an open question whether RA-MSP admits an $O(1)$-competitive algorithm even without knowing the matroid upfront. In this work, we answer this question affirmatively. Our result makes RA-MSP the first well-known MSP variant with an $O(1)$-competitive algorithm that does not need to know the underlying matroid upfront and without any restriction on the underlying matroid. Our approach is based on first approximately learning the rank-density curve of the matroid, which we then exploit algorithmically.

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arXiv2023

Congruency-Constrained TU Problems Beyond the Bimodular Case

A long-standing open question in Integer Programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs $\min\{c^\top x\colon\ Tx\leq b,\ γ^\top x\equiv r\pmod{m},\ x\in\mathbb{Z}^n\}$ with a totally unimodular constraint matrix $T$. Such problems have been shown to be polynomial-time solvable for $m=2$, which led to an efficient algorithm for integer programs with bimodular constraint matrices, i.e., full-rank matrices whose $n\times n$ subdeterminants are bounded by two in absolute value. Whereas these advances heavily relied on existing results on well-known combinatorial problems with parity constraints, new approaches are needed beyond the bimodular case, i.e., for $m>2$. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for $m=3$. Furthermore, for general $m$, our techniques also allow for identifying flat directions of infeasible problems, and deducing bounds on the proximity between solutions of the problem and its relaxation.

math.OC↗

arXiv2023

Advances on Strictly $Δ$-Modular IPs

There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant $Δ\in \mathbb{Z}_{>0}$, $Δ$-modular IPs are efficiently solvable, which are IPs where the constraint matrix $A\in \mathbb{Z}^{m\times n}$ has full column rank and all $n\times n$ minors of $A$ are within $\{-Δ, \dots, Δ\}$. Previous progress on this question, in particular for $Δ=2$, relies on algorithms that solve an important special case, namely strictly $Δ$-modular IPs, which further restrict the $n\times n$ minors of $A$ to be within $\{-Δ, 0, Δ\}$. Even for $Δ=2$, such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly $Δ$-modular IPs. Prior advances were restricted to prime $Δ$, which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly $Δ$-modular IPs in strongly polynomial time if $Δ\leq4$.

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arXiv2023

A Simple Optimal Contention Resolution Scheme for Uniform Matroids

Contention resolution schemes (or CR schemes), introduced by Chekuri, Vondrak and Zenklusen, are a class of randomized rounding algorithms for converting a fractional solution to a relaxation for a down-closed constraint family into an integer solution. A CR scheme takes a fractional point $x$ in a relaxation polytope, rounds each coordinate $x_i$ independently to get a possibly non-feasible set, and then drops some elements in order to satisfy the constraints. Intuitively, a contention resolution scheme is $c$-balanced if every element $i$ is selected with probability at least $c \cdot x_i$. It is known that general matroids admit a $(1-1/e)$-balanced CR scheme, and that this is (asymptotically) optimal. This is in particular true for the special case of uniform matroids of rank one. In this work, we provide a simple and explicit monotone CR scheme for uniform matroids of rank $k$ on $n$ elements with a balancedness of $1 - \binom{n}{k}\:\left(1-\frac{k}{n}\right)^{n+1-k}\:\left(\frac{k}{n}\right)^k$, and show that this is optimal. As $n$ grows, this expression converges from above to $1 - e^{-k}k^k/k!$. While this asymptotic bound can be obtained by combining previously known results, these require defining an exponential-sized linear program, as well as using random sampling and the ellipsoid algorithm. Our procedure, on the other hand, has the advantage of being simple and explicit. This scheme extends naturally into an optimal CR scheme for partition matroids.

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arXiv2022

Simple Random Order Contention Resolution for Graphic Matroids with Almost no Prior Information

Random order online contention resolution schemes (ROCRS) are structured online rounding algorithms with numerous applications and links to other well-known online selection problems, like the matroid secretary conjecture. We are interested in ROCRS subject to a matroid constraint, which is among the most studied constraint families. Previous ROCRS required to know upfront the full fractional point to be rounded as well as the matroid. It is unclear to what extent this is necessary. Fu, Lu, Tang, Turkieltaub, Wu, Wu, and Zhang (SOSA 2022) shed some light on this question by proving that no strong (constant-selectable) online or even offline contention resolution scheme exists if the fractional point is unknown, not even for graphic matroids. In contrast, we show, in a setting with slightly more knowledge and where the fractional point reveals one by one, that there is hope to obtain strong ROCRS by providing a simple constant-selectable ROCRS for graphic matroids that only requires to know the size of the ground set in advance. Moreover, our procedure holds in the more general adversarial order with a sample setting, where, after sampling a random constant fraction of the elements, all remaining (non-sampled) elements may come in adversarial order.

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arXiv2021

New Approximations and Hardness Results for Submodular Partitioning Problems

We consider the following class of submodular k-multiway partitioning problems: (Sub-$k$-MP) $\min \sum_{i=1}^k f(S_i): S_1 \uplus S_2 \uplus \cdots \uplus S_k = V \mbox{ and } S_i \neq \emptyset \mbox{ for all }i\in [k]$. Here $f$ is a non-negative submodular function, and $\uplus$ denotes the union of disjoint sets. Hence the goal is to partition $V$ into $k$ non-empty sets $S_1,S_2,\ldots,S_k$ such that $\sum_{i=1}^k f(S_i)$ is minimized. These problems were introduced by Zhao et al. partly motivated by applications to network reliability analysis, VLSI design, hypergraph cut, and other partitioning problems. In this work we revisit this class of problems and shed some light onto their hardness of approximation in the value oracle model. We provide new unconditional hardness results for Sub-$k$-MP in the special settings where the function $f$ is either monotone or symmetric. For symmetric functions we show that given any $ε> 0$, any algorithm achieving a $(2 - ε)$-approximation requires exponentially many queries in the value oracle model. For monotone objectives we show that given any $ε> 0$, any algorithm achieving a $(4/3 - ε)$-approximation requires exponentially many queries in the value oracle model. We then extend Sub-$k$-MP to a larger class of partitioning problems, where the functions $f_i(S_i)$ can be different, and there is a more general partitioning constraint $ S_1 \uplus S_2 \uplus \cdots \uplus S_k \in \mathcal{F}$ for some family $\mathcal{F} \subseteq 2^V$ of feasible sets. We provide a black box reduction that allows us to leverage several existing results from the literature; leading to new approximations for this class of problems.

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arXiv2020

Weakly Submodular Function Maximization Using Local Submodularity Ratio

Weak submodularity is a natural relaxation of the diminishing return property, which is equivalent to submodularity. Weak submodularity has been used to show that many (monotone) functions that arise in practice can be efficiently maximized with provable guarantees. In this work we introduce two natural generalizations of weak submodularity for non-monotone functions. We show that an efficient randomized greedy algorithm has provable approximation guarantees for maximizing these functions subject to a cardinality constraint. We then provide a more refined analysis that takes into account that the weak submodularity parameter may change (sometimes improving) throughout the execution of the algorithm. This leads to improved approximation guarantees in some settings. We provide applications of our results for monotone and non-monotone maximization problems.

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arXiv2020

A Parameterized Family of Meta-Submodular Functions

Submodular function maximization has found a wealth of new applications in machine learning models during the past years. The related supermodular maximization models (submodular minimization) also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints. Hence, while there are well-developed tools for maximizing a submodular function subject to a matroid constraint, there is much less work on the corresponding supermodular maximization problems. We give a broad parameterized family of monotone functions which includes submodular functions and a class of supermodular functions containing diversity functions. Functions in this parameterized family are called \emph{$γ$-meta-submodular}. We develop local search algorithms with approximation factors that depend only on the parameter $γ$. We show that the $γ$-meta-submodular families include well-known classes of functions such as meta-submodular functions ($γ=0$), metric diversity functions and proportionally submodular functions (both with $γ=1$), diversity functions based on negative-type distances or Jensen-Shannon divergence (both with $γ=2$), and $σ$-semi metric diversity functions ($γ= σ$).

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arXiv2020

Beyond Submodular Maximization via One-Sided Smoothness

The multilinear framework has achieved the breakthrough $1-1/e$ approximation for maximizing a monotone submodular function subject to a matroid constraint. This framework has a continuous optimization part and a rounding part. We extend both parts to a wider array of problems. In particular, we make a conceptual contribution by identifying a family of parameterized functions. As a running example we focus on solving diversity problems $\max f(S)=\frac{1}{2}\sum_{i,j\in A}A_{ij}:S\in\mathcal{M}$, where $\mathcal{M}$ is a matroid. These diversity functions have $A_{ij}\geq 0$ as a measure of dissimilarity of $i,j$, and $A$ has $0$-diagonal. The multilinear framework cannot be directly applied to the multilinear extension of such functions. We introduce a new parameter for functions $F\in{\bf C}^2$ which measures the approximability of the associated problem $\max\{F(x):x\in P\}$, for solvable downwards-closed polytopes $P$. A function $F$ is called one-sided $σ$-smooth if $\frac{1}{2}u^T\nabla^2 F(x) u\leqσ\cdot\frac{||u||_1}{||x||_1}u^T\nabla F(x)$ for all $u,x\geq 0$, $x\neq 0$. We give an $Ω(1/σ)$-approximation for the maximization problem of monotone, normalized one-sided $σ$-smooth $F$ with an additional property: non-positive third order partial derivatives. Using the multilinear framework and new matroid rounding techniques for quadratic objectives, we give an $Ω(1/σ^{3/2})$-approximation for maximizing a $σ$-semi-metric diversity function subject to matroid constraint. This improves upon the previous best bound of $Ω(1/σ^2)$ and we give evidence that it may be tight. For general one-sided smooth functions, we show the continuous process gives an $Ω(1/3^{2σ})$-approximation, independent of $n$. In this setting, by discretizing, we present a poly-time algorithm for multilinear one-sided $σ$-smooth functions.

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arXiv2019

Multivariate Submodular Optimization

Submodular functions have found a wealth of new applications in data science and machine learning models in recent years. This has been coupled with many algorithmic advances in the area of submodular optimization: (SO) $\min/\max~f(S): S \in \mathcal{F}$, where $\mathcal{F}$ is a given family of feasible sets over a ground set $V$ and $f:2^V \rightarrow \mathbb{R}$ is submodular. In this work we focus on a more general class of \emph{multivariate submodular optimization} (MVSO) problems: $\min/\max~f (S_1,S_2,\ldots,S_k): S_1 \uplus S_2 \uplus \cdots \uplus S_k \in \mathcal{F}$. Here we use $\uplus$ to denote disjoint union and hence this model is attractive where resources are being allocated across $k$ agents, who share a `joint' multivariate nonnegative objective $f(S_1,S_2,\ldots,S_k)$ that captures some type of submodularity (i.e. diminishing returns) property. We provide some explicit examples and potential applications for this new framework. For maximization, we show that practical algorithms such as accelerated greedy variants and distributed algorithms achieve good approximation guarantees for very general families (such as matroids and $p$-systems). For arbitrary families, we show that monotone (resp. nonmonotone) MVSO admits an $α(1-1/e)$ (resp. $α\cdot 0.385$) approximation whenever monotone (resp. nonmonotone) SO admits an $α$-approximation over the multilinear formulation. This substantially expands the family of tractable models for submodular maximization. For minimization, we show that if SO admits a $β$-approximation over \emph{modular} functions, then MVSO admits a $\frac{β\cdot n}{1+(n-1)(1-c)}$-approximation where $c\in [0,1]$ denotes the curvature of $f$, and this is essentially tight. Finally, we prove that MVSO has an $αk$-approximation whenever SO admits an $α$-approximation over the convex formulation.

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arXiv2018

Multi-Agent Submodular Optimization

Recent years have seen many algorithmic advances in the area of submodular optimization: (SO) $\min/\max~f(S): S \in \mathcal{F}$, where $\mathcal{F}$ is a given family of feasible sets over a ground set $V$ and $f:2^V \rightarrow \mathbb{R}$ is submodular. This progress has been coupled with a wealth of new applications for these models. Our focus is on a more general class of \emph{multi-agent submodular optimization} (MASO) which was introduced by Goel et al. in the minimization setting: $\min \sum_i f_i(S_i): S_1 \uplus S_2 \uplus \cdots \uplus S_k \in \mathcal{F}$. Here we use $\uplus$ to denote disjoint union and hence this model is attractive where resources are being allocated across $k$ agents, each with its own submodular cost function $f_i()$. In this paper we explore the extent to which the approximability of the multi-agent problems are linked to their single-agent {\em primitives}, referred to informally as the {\em multi-agent gap}. We present different reductions that transform a multi-agent problem into a single-agent one. For maximization we show that (MASO) admits an $O(α)$-approximation whenever (SO) admits an $α$-approximation over the multilinear formulation, and thus substantially expanding the family of tractable models. We also discuss several family classes (such as spanning trees, matroids, and $p$-systems) that have a provable multi-agent gap of 1. In the minimization setting we show that (MASO) has an $O(α\cdot \min \{k, \log^2 (n)\})$-approximation whenever (SO) admits an $α$-approximation over the convex formulation. In addition, we discuss the class of "bounded blocker" families where there is a provably tight O$(\log n)$ gap between (MASO) and (SO).

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